
// Make newform 325.6.a.f in Magma, downloaded from the LMFDB on 26 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_325_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_325_6_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_325_6_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-47440, -10688, 6584, 328, -161, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-4800, -2956, 212, 125, -2, -1], [-8400, 3196, 28, -125, 2, 1], [-2260, -808, 125, 10, -1, 0], [1288, -116, -101, 2, 1, 0]];
    Rf_basisdens := [1, 1, 240, 240, 60, 12];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_325_a();"
function MakeCharacter_325_a()
    N := 325;
    order := 1;
    char_gens := [27, 301];
    v := [1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_325_a_Hecke(Kf)
    return MakeCharacter_325_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 6;
    raw_aps := [[0, -1, 0, 0, 0, 0], [-3, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0], [-27, -6, 0, 0, -1, 1], [124, 28, 1, -8, -4, 1], [169, 0, 0, 0, 0, 0], [-734, -6, -12, 24, -19, -2], [948, -124, 63, 24, 20, -5], [205, -6, 37, 40, -43, -2], [-602, -94, -14, -32, 9, 54], [2892, -302, -1, -16, -53, -43], [-984, 640, -74, -104, -108, -6], [2122, 6, 108, -168, 187, -46], [-475, 176, -539, -368, -48, -48], [-69, -338, -392, -224, 229, 47], [3114, -962, 556, -56, 471, -18], [-1812, -96, 505, -536, 46, 89], [10126, -1270, 622, -80, -67, 130], [2645, 3886, -938, 16, -27, -45], [35758, -466, -35, -80, -575, 209], [-8838, -2352, 1474, -632, -8, -64], [4674, 2232, -1518, -912, 828, 240], [-4567, -2678, 112, 208, 263, -851], [-10158, 2192, -2432, 1648, 608, -164], [-20954, 2844, 4920, 288, 590, 280], [15810, 11780, -968, -1424, 834, 684], [10153, -2846, 1319, 2072, -963, -456], [33727, -4604, -3099, -384, -1538, -934], [52402, -2148, 116, 656, -2002, 364], [62290, -3850, 2668, -1640, 2259, 30], [-50121, 4818, -1259, -2648, 4997, 2344], [12966, 2084, 6184, 3328, -4330, -782], [-8966, 23240, 2480, -2320, 4132, 1292], [-49474, 16484, 1906, -512, 4418, 412], [-63038, 5340, 960, -5088, -1586, 1796], [-44830, 32434, -6921, -672, -4009, 1003], [10794, 20604, -4684, -6928, -2626, -1196], [-116441, 4486, 2858, 2048, 6061, -3091], [20657, 24294, 6500, 5648, -1355, 173], [278838, -3014, 12308, 4616, -3099, -942], [199010, -28608, 4140, -672, 3284, 2662], [52062, -13082, 10194, 4368, -12853, -3422], [-76608, -40480, 13932, 2160, 4232, -3620], [-152018, -22644, -2184, 2304, -4402, 4444], [363366, 42732, 264, 7344, -3594, 132], [192310, -21976, -11408, 3904, -4768, -4358], [2054, -10736, -27424, -5104, -2308, 2494], [-83259, -53222, -21848, -8192, 7167, 2409], [-188737, 68394, 29070, 3216, 9847, -1171], [-147294, 27218, 10396, 14296, 6265, 866], [383814, -21036, -39684, -3024, 2210, 4096], [478766, 23290, -18103, -3424, 17279, 4453], [-417498, 95450, 12960, -4248, -2507, -3058], [-277928, -15852, -20654, 23632, 16910, -398], [893910, 36960, 2796, 27504, -512, -9424], [4645, 3718, -17015, 27640, -28877, 98], [824634, -117200, 13032, 3936, -9920, -11200], [114506, 71634, 3523, -4400, -449, 12887], [-157582, -69370, -7716, -30120, 1619, 7978], [-475002, -108490, -200, 24664, -10605, -486], [-508259, 125556, -36683, -12512, -7694, -2932], [680536, -73924, -26662, -18664, 19954, 5966], [-659525, -89594, 8284, 5584, -5983, 5563], [1065798, -40180, -662, -4112, 5486, -1856], [928482, 95290, -41960, -1064, 2725, 7562], [-71344, 5592, 5970, 34344, -11296, -1862], [688476, -39992, -17653, 9128, -642, -10161], [481726, -8222, 12652, -5720, 44417, 2614], [136527, 216700, -51497, -7376, 12438, -11904], [562678, -68546, -14632, 32024, 15895, 6362], [764966, 112320, -48490, -13864, 1672, -23740], [867728, -11194, -51123, -5568, -783, 11391], [-1041429, 21422, -70985, -75368, 14599, 5210], [1087066, 68532, -6028, 21392, -41366, 22172], [797834, -15768, -22367, 85912, -3634, -13433], [251913, 232006, 62108, 2864, -3483, 16533], [860750, -249856, -108432, 11808, -48840, -5016], [-776744, -305164, -38802, -28344, -20178, 3342], [-639010, -179072, -29672, -21296, 54752, 1024], [954450, -104390, -46476, 10728, -76811, 1382], [179964, 173020, -44450, -11456, -62126, 14810], [-175230, 101372, -224664, -4368, 36886, -13720], [487896, -129882, -38517, 39024, -47467, -4295], [900578, -272248, -45980, 3232, -15948, 25848], [1747784, -246220, 163304, 54608, 20970, 6612], [-655103, 66564, -49475, 27136, 10386, 30276], [1061858, -235610, -8380, -14200, -12397, -13682], [480090, -417496, -67720, -15472, 17916, -7080], [-3000222, -446138, -64108, -59368, 51451, 15386], [1128437, -65990, 99766, -62000, -39101, -1993], [-6541, 329560, -69343, -142432, -10056, -19086], [2424348, 30102, -187347, -78432, 14145, 27675], [-925155, -171838, 5022, 75888, 44935, 41903], [-1017584, -161560, -117170, -24128, -66368, 3866], [-1983336, -418328, 109447, -34856, -17378, 21011], [-66315, -244858, -144901, -82216, -26417, 54128], [519594, 449374, 180984, -3240, 32479, 30158], [-1525772, -303558, 233778, -57456, 29281, 39176], [2166945, -544500, -11519, -29792, 70606, 8264], [2116302, 489066, 243192, 128856, -79067, 14786], [3376739, -551872, -3527, 93856, 6236, -11834], [-1098592, 322776, 21158, -67144, -109440, -20118], [1017609, 12352, 251789, 18080, 70240, 5300], [1647568, -1222614, -143414, -105488, -28047, 13152], [-3865754, -219236, -303302, 20656, -127314, -25884], [485250, -450992, -10166, 36808, -22376, -92992], [3408951, -115170, -155454, 179232, -45463, -19307], [2788810, -178656, 184208, 107600, 103576, -53608], [5836290, 687572, -119058, -21888, 22546, -28912], [-3253182, 835432, 174312, 75072, -96564, -93384], [-94953, -394154, 170901, -16200, 13943, -31400], [-6222602, 855392, 152040, 170304, -11688, 11076], [-2991858, 867156, -227232, -200928, 20850, 42348], [6510264, 218912, -304375, 79160, 8310, -24075], [4312216, 119850, 55557, 80064, 16371, 70095], [-6491110, -245548, 10704, 177168, -77070, -27708], [-112239, 1668950, 215980, 141952, -37435, -32387], [-3361657, 1215086, -49591, -3688, 48907, 51824], [-151962, -260496, 77524, -177584, 129304, -8704], [169560, 316292, 125546, -266368, 216694, -28630], [5596942, -596724, 103920, -115728, 203462, 42388], [-7097602, -874666, -95292, -6168, 6211, -2026], [-4588474, -674154, -5836, 89960, 152827, -63910], [7957893, -1413626, -28858, -228976, 545, -44573], [131956, 827296, 87359, 74840, 144210, -28821], [2965090, -436824, 341040, 112800, -166948, -64976], [11445790, -140078, 502340, 67352, 68289, 34758], [-451242, 682252, -221464, -172336, 47458, -63346], [-8508969, -473306, -91709, 231496, -67309, 4942], [-2631254, 254832, 44088, -190752, 239072, 1372], [-9354150, 1013352, -117637, -8968, -218086, -57671], [551625, -820482, -25192, 100688, -323335, -40223], [6645864, -684704, -381794, 95296, 114084, 126678], [687222, 1503918, 173164, 24088, -207585, 42342], [1882374, 1555024, 275696, -74560, 204864, 66516], [-447354, 1194568, 19260, -114240, 57188, 100264], [6369664, -1111652, -269730, 164040, 18898, 62990], [-2570069, 1195182, -673024, 163568, -77539, 72247], [11063130, -604236, 122632, 70384, 97618, 60224], [-3473636, 811282, -324274, 331136, 74477, 56272], [5456640, -800308, -64085, -225992, 391512, -11505], [-1918702, -334420, 542580, 403296, 99966, -113280], [-5177783, -340994, -127897, 296024, -171813, -111420], [21466909, 730922, 375544, -104336, -355425, 61353], [1985994, 494150, -567026, -11552, -439741, 96874], [10609890, 911906, 3987, 424032, -488249, -29581], [-6424092, 339720, -1051014, 37896, -258144, -39690], [-4138982, 586008, -330012, -314064, 28852, -115324], [-3544728, 1019112, -586030, 237008, 152776, 14894], [13130829, 65202, 8620, -127712, 65635, -68611], [-3210054, -98120, -501856, 482336, -362676, -32784], [-354168, 698990, 96710, 193424, -109197, -4980], [-18089363, 273696, 71325, 175104, 183864, 125400], [2991723, -1974342, -16653, 396264, 380021, 119698], [13973007, -37096, 291505, -310208, -348928, 51070], [8863206, -101424, 343720, 23536, -150764, 212618], [-13717934, -245044, -249542, -162944, 52262, -182972], [6910950, 1737438, -670636, -331576, -36249, -188406], [1574118, 3153912, 45588, -194400, 67572, -136812], [14455954, 365358, 571836, -280728, 490951, 176366], [9711765, -476854, -697220, 9520, -240057, 58197], [4947522, 31900, -449660, -238592, 29126, 122128], [-14796011, 2263790, -244034, -238208, -96835, 155851], [-1641358, 260904, -855628, -172336, 61656, -158106], [287778, 2098032, -767326, -309400, -109384, 384952], [14116329, 2299042, -333898, -779440, -456593, 41387], [3135188, 1772252, 284272, -203264, -155098, 126640], [10817478, 115386, -1343676, -515400, -112531, 98782], [-16531562, 2356224, 1040012, 387584, 22816, -169672], [-9354078, -530230, 159284, 504440, -541251, -192582], [3387974, -2727764, 792541, -289256, 35792, -14285], [-3265366, -3487254, -330176, 541528, -465587, 29174], [-3305080, -538036, 896298, 282480, 198498, 218430], [5953462, -2198400, 218878, -293672, -424248, 90372], [-671532, 1233124, 523636, 206608, 644290, 118844], [-14696294, -384840, -215840, -12992, -54564, 243168], [20800830, 1076900, 667495, 240376, -360184, -194039], [-6835722, -3284382, 1279512, 624648, 2729, -371282], [-11103423, -2392682, -480419, -254840, 403403, 226606], [7204470, -1836784, -1319064, 303360, -471504, -145656], [-11051981, -3504154, 921952, 776704, 186113, 18319], [-4327030, 314656, 593790, 829728, 442576, 117368], [7176530, 338482, -1363864, -419992, 318385, 188114], [-15577322, 1986704, 1062192, 522480, -475312, -268568], [-21746009, 3039690, 724138, 253504, -101977, -214199], [20590170, 1505826, -464402, -502352, -479567, 130346], [2275110, -1924776, 271260, -415776, 164388, -163500], [-1773143, -2018514, 242672, -116992, 724073, -169463], [6925190, 311302, -457924, 498392, 386035, -55882], [43722206, -727978, -753809, -530144, -1439, -295261], [7261254, 800262, 772804, 212728, -1344781, -104606], [14634077, 76650, 490876, -184112, 196023, 123261], [16904978, -658488, 195404, 251888, 784760, 70726], [6606374, -3944016, -1736400, -705456, -140944, -59444], [3774303, -5032164, 656883, -190512, 232062, 634296], [-24509838, 1727984, 298124, -96928, -718968, -573624], [-15877706, 732546, -708684, -209016, -515311, 214810], [36881586, 1212332, -361520, -218480, -361042, -306848], [9197926, -1463940, -1081346, 75496, -93322, -455432], [27902217, -3535794, -158381, -883208, -355637, 36452], [28086798, 1937150, -1031432, -1140008, 956471, 163426], [23521902, 2305684, 1347860, 1687904, -500434, -214070], [5279014, 3876920, -1008640, -726784, 641268, -12156], [31574076, -1055894, -779594, -1133984, -334311, -318600], [-5436638, -1747908, 76255, -62408, -1012132, -44639], [-18593314, -429770, -2679452, -983144, 52643, 55246], [5006046, -1597814, -963147, 183360, 276239, 486145], [-11555427, 320596, -1254971, 193072, -105038, 382832], [29445658, -5081116, 370100, 365744, -134134, -473936], [-25562768, -447784, -1859262, -250560, -202544, -397162], [43769922, -2778112, -1131132, -94368, 430304, 75328], [-3343910, 4515442, 533342, -387952, -456727, -504458], [5589777, 4618806, -678957, 355272, 897391, -206356], [-38975441, -25806, -2234532, -684960, -300689, -296989], [17541428, -2357886, 1426127, 221840, 530515, 49889], [-20804446, 3794350, 1168064, -1813336, -19265, 252050], [-15699141, -1737446, 3308547, 267000, -555819, -241926], [-9383346, -1677928, -36992, -1193120, -370804, -289112], [40914091, 2048926, 493918, -135152, 936337, -159955], [3999134, 2000888, -1204440, -242304, 593196, 343248], [-45238102, 2420482, 175902, -45648, 1573217, -36746], [39238846, -6240410, 627631, -397232, 575129, 34639], [28059910, 2390240, -1558744, 407120, -1230272, -8344], [-22623929, -5610974, -827840, 884944, -826873, -445049], [32092403, 5902420, 558653, 1319600, -871890, 37674], [10763590, 7734974, 2074542, -45312, 2403127, 53078], [1666710, -514172, 2533160, 2103104, 475322, -277064], [11775804, 2564032, -815930, -734672, 693836, -15590], [20592943, -5768554, 795195, -1393128, -228501, -2178], [59949094, -2256216, -660053, 432040, -612798, 244137], [57059358, -360844, 3421876, 1264576, 1703162, -13148], [-38130936, 2666648, -317178, -242928, -1563752, -467470], [33244678, 11942894, 420249, -427968, 786649, 78317], [-22205434, 1205452, -1860520, -779440, -670354, 519148], [12057153, -1888712, 572953, 1465936, 1192164, -260340], [15097653, 284218, 1284986, 431504, -1457973, -51021], [-13589670, 3463646, 810112, 992056, 922943, 670366], [-17020332, 1553416, -2713326, -2546328, 42912, -163458], [11855978, 7820224, -1547648, 236224, 1186156, -765742], [-5299268, -6815548, 3018620, 842624, -596638, -106820], [37381971, 854294, -659024, -139856, 49457, -87785], [-3934214, 7540224, -2330, -14528, -584688, 472152], [-25522371, 2048058, -2396609, 1256776, -1036703, 297968], [-2076610, -5702870, -2811980, -2464520, -837171, 293862], [47483554, -7339908, 3231664, -1043552, 1714334, -21656], [13181244, -2647732, -268996, -244048, -1624714, -42212], [27396063, 4542934, 2116300, -329696, 1491993, 134079], [34692628, -5848416, 1418321, -1583368, 1505950, -358051], [12623898, -7308240, -2743823, 1174024, -2048126, -1020529], [22602997, 4927154, 1412899, -1415192, -1056011, 415976], [-16607656, -10545036, 867702, 1053576, -133266, 123222], [-76110786, -2214592, 1790504, 960224, 1110168, -700764], [-30277923, 2895642, 1182444, -86112, 1782039, -1267275], [-18971798, 10220464, 1974000, -1222080, 915040, -292048], [34089394, -2265972, -6326752, -2612224, 145494, 134988], [3988640, 9541768, -3195551, 130696, -920822, -827227], [-43535890, -1763602, -3269458, 538976, -49089, -1067238], [17202829, 1666922, 409870, -796832, 1834819, 968903], [-52067106, 13012924, -2148688, 930080, -1074346, -87632], [5713174, -6964364, 405218, -249736, -852206, 293624], [10126441, 5118130, 1982401, -180008, -1660383, -173634], [79013707, 5740184, -4133157, -963648, -471276, 933612], [-11679862, 4805858, -2904020, 696616, -1565719, 715078], [-1154262, 3676754, -3706736, -2007944, -513127, 295558], [-44958534, 6812964, 496998, -1418232, 202290, 806076], [24280358, -6241536, -3605762, 2333872, 106648, -656092], [14578354, -7318694, 1915552, 2436568, 1560821, 1016950], [4126682, -2293440, -3578996, -1216736, -703560, 649164], [82349367, 3597730, -1925642, -980144, 1661939, 844045], [-94628522, 2687010, -1418832, 1654872, 390553, 576794], [-81209582, 3590444, -2153256, -235344, -31706, 444668], [12157903, -1146010, -4376334, -923424, -311891, 1081421], [23410710, 851484, 3140192, 3858464, -272058, -150396], [-4374502, 6105320, -927720, -209184, 962456, 1280086], [-77647706, -3771598, -1048744, 1664120, 1420665, 609366], [-145656919, 4653110, 3104120, 1701296, -533211, -18159], [41905079, 3311570, 658542, 3211344, -2975397, -523395], [26156042, -5718698, -3720248, -1292312, 54387, -446694], [-73189222, -6676778, 4303468, 1909192, -541973, -1181794], [41434010, -12662204, 656530, 514384, -1622342, 6248], [51266393, -2352662, -1304089, -1128472, 668049, 787644], [-57001064, -5562458, 4590389, 735584, -757487, 203963], [74768411, 10519870, 1986773, -662872, 2692451, 675748], [-91641590, 9680866, -3501120, 674808, 295817, -194594], [-27868153, -28387042, -239328, 353808, 1178717, -64061], [12842360, 5065832, 621390, 3133536, -496728, -205338], [6760070, 13683220, -2396932, -2304976, -1379614, 293512], [-43946994, -1185204, -3508416, 131472, -933122, 1097096], [-4748488, 1781582, -261239, 419392, 889709, 580771], [33759184, -5853854, 2885298, 3073296, 304909, 509768], [64659498, 11630154, -2587522, 992432, -772363, -165746], [-25268321, 1807324, 171997, 910000, 1061738, -118070], [63847934, -9696712, 3193704, 821136, -1724004, -656244], [57988692, -4474240, 8341009, 1878904, 328926, -447723], [109637658, -4774012, 850776, -374208, 1793698, 13772], [28742582, 941776, 4247344, 1263040, -1423696, 298600], [66871502, 19497136, -2695180, -719824, 1067852, -1424666], [-116836674, -3469484, 450932, 3274832, -221246, -203632], [-118309822, 8650320, -3143182, 1625552, 85888, 1133852], [-26960147, -650310, -4973568, -3333072, -1397337, 269433], [-15390638, 7578284, 3440120, -1038640, -1979730, -147696], [14669858, -10711140, 1675576, -1005872, 193310, -1835948], [-82476858, -19095856, 3120972, -1938528, 1242604, 71726], [-50369791, -1924560, 1722425, -3582544, -1953968, -59716], [-120952808, 9075644, -416048, 1795840, 2464646, 52168], [-79032066, 1879300, 1896538, -116744, 2200298, -984320], [-842821, -207320, -4271855, 443008, 2601856, 1348106], [111289306, -8153940, 6144320, 1879568, 1296998, -621596], [-74653490, 8494038, 3125899, -1267040, -1967799, -107757], [57835390, 19617752, -4728232, -3704368, 1909668, 58068], [-163093803, -9670334, 3910997, -1754008, 242457, 1062090], [-24426702, -5770314, -6123048, -1528344, -1869213, -977838], [-33005208, 11541698, 78514, -4441232, 379341, 1997700], [30745869, -10981472, -2476745, 4627744, -1212444, -1668870], [147666829, -11938434, -5823088, 6395888, -1518303, -944763], [-162696592, -11567030, -1173978, 292608, 562913, -628316], [19562956, 9564988, 305643, -2738280, 2417544, -668577], [142004326, -8358378, -5122007, 1290976, 415373, -21851], [-24032586, 14892516, -4833510, -2375880, 2536266, 376296], [-15511242, 4022464, 3493208, -5969248, 199464, 103296], [-157970296, 1369764, -2064546, -1517712, 330382, -348682], [68485654, 610240, 3434186, 81800, -4319384, -140284], [25516410, -15480998, -4424058, 1412880, -1691267, 675230], [4646837, -7392974, 3520754, -519376, 1820695, 722291], [-32726662, -13965544, -1601022, 3498504, -196612, -187064], [-21727632, 5054694, 4904402, -209056, 3222911, 1439332], [-18450976, -5074688, 3771041, 1236680, 2037926, -1346075], [-152214311, 1871610, -47764, 3140720, 792263, 54985], [94735335, -2773682, 5852681, -3522088, 7104267, 754992], [-71750498, -11583956, -4179028, 1124192, -3268714, -193436], [-10740054, -20532208, -3209600, -2064416, -1316704, 510364], [96016494, -4171720, 8153472, 2170992, -3048988, -51476], [7904046, 4620460, -3573762, 1667040, -794042, -2320612], [49101711, 771018, 1910542, 862864, -4130729, 683885], [57466066, 9766328, -534003, -1297128, 2417502, 1595847], [-77355591, 6356478, -3077030, -3199376, -656243, 2366555], [-109626298, -4941192, -939180, 2367696, 3411340, -942088], [-53270430, -7494988, 7757648, -5141536, -655134, 576180], [81904978, -12269934, -1810216, -5350744, -2255287, -491462], [25421815, -7349230, -285967, 3345752, 1152237, 982404], [43618858, 4722732, 4451528, 3075056, 3736174, 1282136], [-25803910, -610674, 10467132, 2982936, 2105087, 1644262], [-186013966, 2965476, 3554432, -727360, -1154942, -1716916], [8257422, 22922520, 6219762, -3273024, 4481092, 267884], [-66186902, 34367516, -3387488, 312784, -1858106, -1612408], [-234157550, -12446268, -1601060, -2271824, 2111142, 600978], [-60980478, -13026736, -3963208, 4114208, -1988480, 1203512], [243546709, -7485810, -1977604, -763360, -918151, -894695], [-127069902, 8174178, -7035765, -3794880, -3258633, 1592751], [63450938, -15951952, 10138252, 133264, -1599920, -1717180], [72355608, -4632032, 380543, 5450216, -7766390, -615145], [76056690, -27306196, 3589720, -2050016, -1370986, 868168], [109350670, 1242404, -2130128, 2788576, -316182, 1608792], [100459737, 655678, -3102597, -2646888, 3667587, -1416240], [26960982, -24727928, 13037984, 6237248, -1948060, -2926496], [-225694350, -1002102, -5466504, -242232, 3541557, 226854], [-26460466, -18627334, -11306533, -816736, -1111765, -901049], [243723700, -7466896, 4048043, 3181688, -3412558, -1566641], [-44520942, -5064932, -2401336, -3072832, -6381426, 515892], [-12325937, -4256298, 2051400, 4774176, -9689615, -2460157], [-206676804, -15433084, 4429970, 3396632, -2723282, 245498], [122920632, 35603286, -8064050, -2061008, 2377687, -421312], [54117579, 16697970, 372807, 742296, -3795751, -511514], [39134038, 4168364, 12001341, 2975976, -204768, -855213], [-154392413, 8182844, -3027317, -691952, 6573670, 405884], [10297674, -9677090, -4114164, -3942456, 3344607, -1604622], [-34761474, 14415916, -727048, -6714832, 2210174, 1253344], [-61480115, -13088858, -5448066, -6130272, 469049, 978547], [12741538, -4948676, -12212672, -3844784, -50170, -540116], [169658418, -20976736, -14251748, -203120, -1457452, 654742], [233191224, 21187928, 3739108, -5544128, -2010148, 916984], [63635907, -6037134, -7265854, 8911904, -4620589, -348227], [15616946, 10263314, -7065222, 4618272, 6316073, -403178], [-5872422, 5018412, 324144, 1500864, 4822586, -626582], [-26672922, 16178798, -2051560, -2323624, -2368217, -1041382], [-22065944, 21686168, -2944257, 2996616, -2009898, 2032131], [117955758, -45997448, 8209226, -4957696, 4648132, 531860], [-275879334, 3904792, -2403320, 5490736, -5917260, 374952], [-43695604, -46030550, 8097070, -2652224, 149497, 2300708], [115340466, -3201074, 3377748, -5756232, -7696641, -321354], [-11983370, 24415854, 7405668, -663528, 2881303, -1588522], [83878230, -11497828, 9238450, 10063576, -1371610, -2128616], [-285703071, 23159894, -11114848, -2609584, -2772411, 255177], [-21897846, -43346260, -2105254, 1810328, -1114682, -1557556], [325316602, -19276036, 8844095, 1718216, 387852, -1445343], [34492213, -29075966, -769426, -4303888, 8538551, 538231], [230367664, 5103054, -6933325, -4062448, 2342153, 1959649], [-71766078, 14382258, 6068972, 6230696, 4509305, 1984510], [-208211503, 7979418, 905192, 999440, 1334591, -3544763], [-19429337, -3500382, 2235709, 2847640, 406397, -4689716], [76461410, 881306, -10852056, 5983608, -4606171, 1626886], [32538424, 16842464, -18549794, -687848, -1317900, 1084050], [-57440688, -12684512, -1235547, 4423512, 2093326, -18535], [-373474225, -18197228, 4625049, 1705152, -3096562, 1869262], [-116192258, -8606098, 8777571, 5888304, -1237467, -1095705], [514454986, -8206104, -1294884, -490224, -64020, 390660], [269213214, 5436982, -15440175, -5190480, -222051, 1350777], [-141681044, -689154, -8583470, 1059136, 4599011, -1261196], [192809356, -5703676, 1377124, 2616352, -7414774, 395512], [-2417706, -22758724, -12057608, 1678720, -10601418, -864000], [194658318, 11032268, 1739928, 3451392, -5568530, -1003720], [-204449514, -38007762, -1769292, -4755384, 1033279, 2058014], [36171583, -12868778, -6478616, -2095424, -1398639, -183789], [39761062, -31687812, 4038686, -5724376, -3491034, 2830728], [-445285302, -16907104, 3111096, -6277056, 2301944, 2194468], [350150576, 6894316, 5060748, 459936, -947938, 2579632], [189778544, 5156988, 7507426, 6049624, 2930242, 618422], [139297050, -18160964, -4878344, 11136832, -2291098, -369056], [-370545669, 6150846, 1333662, -989760, -5729847, -1971003], [143075352, 32184960, 6068325, 4894920, -5845842, -1040895], [-195006342, -24923436, 1323024, -4505904, 25650, 1587504], [-285633342, 28517386, -6622408, -3458056, 4314405, 2239422], [-181792887, -57564600, 1000775, 901376, 322496, -1126562], [268070408, 12790692, 5879798, -7614640, -3311746, 905134], [10118566, 32337084, 1533696, 6354240, 3673014, -1147848], [-31805938, -18936120, -10197528, -3610080, -1126588, -478520], [-50340046, 24396294, -624683, 5401888, 5688789, -3876039], [-245233953, 31747642, 15789501, -1495608, 1474697, -925316], [72749366, 12127616, 22403262, 10385016, 4278072, 365340], [14435381, 18638602, -2438149, 606152, -10871487, -4093368], [111396486, -6524394, 18923342, -6607072, -2910637, 1385062], [175129854, 64231400, 1062992, -536896, 7037412, -911328], [-162929127, 44769962, -11064422, -2537648, -231973, 2645191], [-91426848, -12164712, 6394858, -9062288, 908928, 2883366], [13298946, -9870968, -6030848, 2869360, 8499804, 3756864], [319744516, -41271792, 12385866, 1670184, 1241860, 410822], [370402065, 7704608, 18178985, 762800, 4342920, 2391504], [49488798, 33045710, -11308088, 1711000, 498095, -5088542], [-29072034, 16314724, 3858219, 3332328, 89264, -4095395], [-92912938, 41126508, 26826, -4259712, -6035378, -2685472]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_325_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_325_6_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_325_6_a_f(:prec:=6)
    chi := MakeCharacter_325_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_325_6_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_325_6_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_325_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-47440, 10688, 6584, -328, -161, 2, 1]>],Snew);
    return Vf;
end function;